Question: Simplify and expand the following expression: $ \dfrac{a + 3}{a + 3}+\dfrac{a - 2}{a - 7} $
Solution: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(a + 3)(a - 7)$ Multiply the first term by $\dfrac{a - 7}{a - 7}$ $ \begin{align*} \dfrac{a + 3}{a + 3} \times \dfrac{a - 7}{a - 7} & = \dfrac{(a + 3)(a - 7)}{(a + 3)(a - 7)} \\ & = \dfrac{a^2 - 4a - 21}{(a + 3)(a - 7)}\end{align*} $ Multiply the second term by $\dfrac{a + 3}{a + 3}$ $ \begin{align*} \dfrac{a - 2}{a - 7} \times \dfrac{a + 3}{a + 3} & = \dfrac{(a - 2)(a + 3)}{(a - 7)(a + 3)} \\ & = \dfrac{a^2 + a - 6}{(a - 7)(a + 3)}\end{align*} $ Now we have: $ = \dfrac{a^2 - 4a - 21}{(a + 3)(a - 7)} + \dfrac{a^2 + a - 6}{(a - 7)(a + 3)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{a^2 - 4a - 21 + a^2 + a - 6}{(a + 3)(a - 7)} $ $ = \dfrac{2a^2 - 3a - 27}{(a + 3)(a - 7)}$ Expand the denominator: $ = \dfrac{2a^2 - 3a - 27}{a^2 - 4a - 21}$